Is it possible add a texture to a plane in the two-dimensional animation editor?
The 2D animation editor actually is the 3D view editor. The "2D Animation" startup file sets the world to white, adds a grease pencil object, and goes into edit mode. Other than that, it's just a different set of layout tabs for Blender.

Go ahead and, in the 2D animation editor tab, orbit around.

In respect to putting an image texture onto a plane, since it is actually the standard 3D space, look for UV and image texture material tutorials.

If you still need help, I might be able to provide a tutorial, but don't worry: There's nothing to be afraid of :)